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%I #58 Dec 26 2017 17:58:46
%S 1,1,1,1,2,1,1,3,3,1,1,4,7,4,1,1,5,9,9,5,1,1,6,13,16,13,6,1,1,7,21,25,
%T 25,21,7,1,1,8,19,36,23,36,19,8,1,1,9,27,49,65,65,49,27,9,1,1,10,49,
%U 64,37,126,37,64,49,10,1,1,11,39,81,125,133,133,125
%N Square array T(n, k) read by antidiagonals, n > 0 and k > 0: for any number n > 0, let f(n) be the function that associates k to the prime(k)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the arithmetic functions with nonnegative integer values and a finite number of nonzero values; let g be the inverse of f; T(n, k) = g(f(n) * f(k)) (where i * j denotes the Dirichlet convolution of i and j).
%C For any n > 0, f(n) corresponds to the function k -> A249344(k, n).
%C For any n > 0 and m > 0, f(n * m) = f(n) + f(m).
%C Also, f(1) = A000004 and f(2) corresponds to k -> A000007(k-1).
%C The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v); as such, f is an homomorphism from the multiplicative group of positive rational numbers to the additive group of arithmetic functions with integer values and a finite number of nonzero values.
%C For any arithmetic function with integer values and a finite number of nonzero values j, g(j) = Product_{k > 0} A000040(k)^j(k).
%C See A296857 for the main diagonal of T.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dirichlet_convolution">Dirichlet convolution</a>.
%F T is completely multiplicative in both parameters:
%F - for any n > 0
%F - and k > 0 with prime factorization Prod_{i > 0} prime(i)^e_i:
%F - T(prime(n), k) = T(k, prime(n)) = Prod_{i > 0} prime(n * i)^e_i.
%F For any m > 0, n > 0 and k > 0:
%F - T(n, k) = T(k, n) (T is commutative),
%F - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F - T(n, 1) = 1 (1 is an absorbing element for T),
%F - T(n, 2) = n (2 is an identity element for T),
%F - T(n, 2^i) = n^i for any i >= 0,
%F - T(n, 4) = n^2 (A000290),
%F - T(n, 8) = n^3 (A000578),
%F - T(n, 3) = A297002(n),
%F - T(n, 3^i) = A297002(n)^i for any i >= 0,
%F - A001221(T(n, k)) <= A001221(n) * A001221(k),
%F - A001222(T(n, k)) = A001222(n) * A001222(k),
%F - A055396(T(n, k)) = A055396(n) * A055396(k),
%F - A061395(T(n, k)) = A061395(n) * A061395(k),
%F - A289508(T(n, k)) = A289508(n) * A289508(k),
%F - T(A000040(n), A000040(k)) = A000040(n * k),
%F - T(A000040(n)^i, A000040(k)^j) = A000040(n * k)^(i * j) for any i >= 0 and j >= 0.
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10
%e ---+-------------------------------------------------
%e 1| 1 1 1 1 1 1 1 1 1 1 -> A000012
%e 2| 1 2 3 4 5 6 7 8 9 10 -> A000027
%e 3| 1 3 7 9 13 21 19 27 49 39 -> A297002
%e 4| 1 4 9 16 25 36 49 64 81 100 -> A000290
%e 5| 1 5 13 25 23 65 37 125 169 115
%e 6| 1 6 21 36 65 126 133 216 441 390
%e 7| 1 7 19 49 37 133 53 343 361 259
%e 8| 1 8 27 64 125 216 343 512 729 1000 -> A000578
%e 9| 1 9 49 81 169 441 361 729 2401 1521
%e 10| 1 10 39 100 115 390 259 1000 1521 1150
%o (PARI) T(n,k) = my(fn=factor(n), pn=apply(primepi,fn[,1]~), fk=factor(k), pk=apply(primepi,fk[,1]~)); prod(i=1, #pn, prod(j=1, #pk, prime(pn[i]*pk[j])^(fn[i,2]*fk[j,2])))
%Y Cf. A000004, A000007, A000012 (first row/column), A000027 (second row/column), A000040, A000290 (fourth row/column), A000578 (eighth row/column), A001221, A001222, A055396, A061395, A249344, A289508, A296857 (main diagonal), A297002 (third row/column).
%K nonn,tabl,mult
%O 1,5
%A _Rémy Sigrist_, Dec 21 2017
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